Problem: $ F = \left[\begin{array}{rr}-2 & 2 \\ 5 & -1 \\ 2 & 5\end{array}\right]$ $ w = \left[\begin{array}{r}-1 \\ 5\end{array}\right]$ What is $ F w$ ?
Because $ F$ has dimensions $(3\times2)$ and $ w$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(3\times1)$ $ F w = \left[\begin{array}{rr}{-2} & {2} \\ {5} & {-1} \\ \color{gray}{2} & \color{gray}{5}\end{array}\right] \left[\begin{array}{r}{-1} \\ {5}\end{array}\right] = \left[\begin{array}{r}? \\ ? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ w$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ w$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ w$ , and so on. Add the products together. $ \left[\begin{array}{r}{-2}\cdot{-1}+{2}\cdot{5} \\ ? \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ w$ and add the products together. $ \left[\begin{array}{r}{-2}\cdot{-1}+{2}\cdot{5} \\ {5}\cdot{-1}+{-1}\cdot{5} \\ ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{-2}\cdot{-1}+{2}\cdot{5} \\ {5}\cdot{-1}+{-1}\cdot{5} \\ \color{gray}{2}\cdot{-1}+\color{gray}{5}\cdot{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}12 \\ -10 \\ 23\end{array}\right] $